On Generic Topological Embeddings

Abstract We show that an embedding of a fixed 0-dimensional compact space K into the Čech–Stone remainder $$\omega ^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ω</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> as a nowhere dense P-set is the unique generic limit, a special object in the category consisting of all continuous maps from K to compact metric spaces. Using categorical Fraïssé theory, we obtain some well-known theorems about Čech–Stone remainders of infinite discrete spaces. Furthermore, we prove several results concerning universality and homogeneity of the space $$\kappa ^\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>κ</mml:mi> <mml:mi>κ</mml:mi> </mml:msup> </mml:math> , with $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> a regular uncountable cardinal, using generalized ultrametrics.